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8 MHR • Chapter 1

Connect English With Mathematics and Graphing Lines

The key to solving many problems in mathematics is the ability to read and understand the words. Then, you can translate the words into mathematics so that you can use one of the methods you know to solve the problem. In this section, you will look at ways to help you move from words to equations using mathematical symbols in order to find a solution to the problem.

� placemat or sheet of paper

Tools

Investigate

How do you translate between words and algebra?

Work in a group of four. Put your desks together so that you have a placemat in front of you and each of you has a section to write on.

1. In the centre of the placemat, write the equation 4x � 6 � 22.

2. On your section of the placemat, write as many word sentences to describe the equation in the centre of the placemat as you can think of in 5 min.

3. At the end of 5 min, see how many different sentences you have among the members of your group.

4. Compare with the other groups. How many different ways did your class find?

5. Turn the placemat over. In the centre, write the expression x � 1.

6. Take a few minutes to write phrases that can be represented by this expression.

7. Compare among the members of your group. Then, check with other groups to see if they have any different phrases.

8. Spend a few minutes talking about what words you used.

9. Reflect Make a list of all the words you can use to represent each of the four operations: addition, subtraction, multiplication, and division.

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Example 1 Translate Words Into Algebra

a) Write the following phrase as a mathematical expression: the value five increased by a number

b) Write the following sentence as a mathematical equation. Half of a value, decreased by seven, is one.

c) Translate the following sentence into an equation, using two variables. Mario’s daily earnings are $80 plus 12% commission on his sales.

Solution

a) Consider the parts of the phrase. • “the value five” means the number 5 • “increased by” means add or the symbol � • “a number” means an unknown number, so choose a variable

such as n to represent the number

The phrase can be represented by the mathematical expression 5 � n.

b) “Half” means

• “of” means multiply • “a value” means a variable such as x • “decreased by” means subtract or � • “seven” is 7 • “is” means equals or � • “one” is 1

The sentence can be represented by the equation x � 7 � 1.

c) Consider the parts of the sentence. • “Mario’s daily earnings” is an unknown and can be represented

by E • “are” means equals or � • “$80” means 80 • “plus” means � • “12% commission on his sales” can be represented by 0.12 � S

The sentence translates into the equation E � 80 � 0.12S.

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1.1 Connect English With Mathematics and Graphing Lines • MHR 9

Sometimes, several sentences need to be translated into algebra. This often happens with word problems.

Example 2 Translate Words Into Algebra to Solve a Problem

Ian owns a small airplane. He pays $50/h for flying time and $300/month for hangar fees at the local airport. If Ian rented the same type of airplane at the local flying club, it would cost him $100/h. How many hours will Ian have to fly each month so that the cost of renting will be the same as the cost of flying his own plane?

10 MHR • Chapter 1

� two or more linear equations that are considered at the same time

linear system

What things are unknown? • the number of flying hours • the total cost

I’ll choose variables for the two unknowns. I will translate the given sentences into two equations. Then, I can graph the two equations and find where they intersect.

The airplane in Example 2 is a Diamond Katana DA40. These planes are built at the Diamond Aircraft plant in London, Ontario.

Did You Know ?

It is a good idea to read a word problem three times.

Read it the first time to get the general idea.

Read it a second time for understanding. Express the problem in your own words.

Read it a third time to plan how to solve the problem.

onnections Literac

Solution

Read the paragraph carefully.

Let C represent the total cost, in dollars. Let t represent the time, in hours, flown.

The first sentence is information that is interesting, but cannot be translated into an equation.

The second sentence can be translated into an equation. Ian pays $50/h for flying time and $300/month for hangar fees at the local airport. C � 50t � 300

The third sentence can also be translated into an equation. If Ian rented the same type of airplane at the local flying club, it would cost him $100/h. C � 100t

The two equations form a . This is a pair of linear relations, or equations, considered at the same time. To solve the linear system is to find the point of intersection of the two lines, or the point that satisfies both equations.

Graph the two lines on the same grid.

linear system

Both equations are in the form y � mx � b. You can use the y-intercept as a starting point and then use the slope to find another point on the graph.

The lines on the graph cross at one point, (6, 600). The is (6, 600).

Check that the solution is correct. If Ian uses his own airplane, the cost is 6 � $50 � $300. This is $600. If he rents the airplane, the cost is 6 � $100. This is $600. So, the solution t � 6 and C � 600 checks.

Write a conclusion to answer the problem.

If Ian flies 6 h per month, the cost will be the same, $600, for both airplanes.

Linear equations are not always set up in the form y � mx � b. Sometimes it is easy to rearrange the equation. Other times, you may wish to graph using intercepts.

Example 3 Find the Point of Intersection

The equations for two lines are x � y � �1 and 2x � y � 2. What are the coordinates of the point of intersection?

Solution

Method 1: Graph Using Slope and y-Intercept

Step 1: Rearrange the equations in the form y � mx � b.

Equation �: x � y � �1

x � y � y � 1 � �1 � y � 1 x � 1 � y

y � x � 1 Equation � becomes y � x � 1. Its slope is 1 and its y-intercept is 1.

Equation �: 2x � y � 2

2x � y � y � 2 � 2 � y � 2 2x � 2 � y

y � 2x � 2 Equation � becomes y � 2x � 2. Its slope is 2 and its y-intercept is �2.

point of intersection

1.1 Connect English With Mathematics and Graphing Lines • MHR 11

� a point where two lines cross

� a point that is common to both lines

point of intersection

0

C

t6

C = 100h

C = 50h + 300 (6, 600)

Rise 100

Rise 50

Run 1

Run 1

7 854321

100

200

300

400

500

600

Step 2: Graph and label the two lines.

Step 3: To check that the point (3, 4) lies on both lines, substitute x � 3 and y � 4 into both original equations.

In x � y � �1: L.S. � x � y R.S. � �1

� 3 � 4 � �1

L.S. � R.S. So, (3, 4) is a point on the line x � y � �1.

In 2x � y � 2: L.S. � 2x � y R.S. � 2

� 2(3) � 4 � 6 � 4 � 2

L.S. � R.S. So, (3, 4) is a point on the line 2x � y � 2.

The solution checks in both equations. The point (3, 4) lies on both lines.

Step 4: Write a conclusion. The coordinates of the point of intersection are (3, 4).

Method 2: Graph Using Intercepts

Step 1: Find the intercepts for each line.

Equation �: x � y � �1

At the x-intercept, y � 0. At the y-intercept, x � 0. x � 0 � �1 0 � y � �1

x � �1 �y � �1 Graph the point (�1, 0). y � 1

Graph the point (0, 1).

Equation �: 2x � y � 2

At the x-intercept, y � 0. At the y-intercept, x � 0. 2x � 0 � 2 2(0) � y � 2

2x � 2 �y � 2 x � 1 y � �2

Graph the point (1, 0). Graph the point (0, �2).

12 MHR • Chapter 1

If I don’t get the same result when I substitute into both equations, I’ve made a mistake somewhere!

0

y

x—2 42

—2

2

4 x — y = 1(3, 4)

2x — y = 2

Step 2: Draw and label the line for each equation.

Step 3: Check by substituting x � 3 and y � 4 into both original equations. See Method 1.

Step 4: Write a conclusion. The coordinates of the point of intersection are (3, 4).

1.1 Connect English With Mathematics and Graphing Lines • MHR 13

0

y

x—2 42

—2

2

4 x — y = —1 (3, 4)

2x — y = 2

Example 4 Solve an Internet Problem

Brian and Catherine want to get Internet access for their home. There are two companies in the area. IT Plus charges a flat rate of $25/month for unlimited use. Techies Inc. charges $10/month plus $1/h for use. If Brian and Catherine expect to use the Internet for approximately 18 h/month, which plan is the better option for them?

Solution

Represent each situation with an equation. Then, graph to see where the two lines intersect to find